Continuous Probability Distributions Chapter 5 Continuous Probability Distributions

Chapter 5 - Chapter Outcomes After studying the material in this chapter, you should be able to: • Discuss the important properties of the normal probability distribution. • Recognize when the normal distribution might apply in a decision-making process.

Chapter 5 - Chapter Outcomes (continued) After studying the material in this chapter, you should be able to: • Calculate probabilities using the normal distribution table and be able to apply the normal distribution in appropriate business situations. • Recognize situations in which the uniform and exponential distributions apply.

Continuous Probability Distributions A discrete random variable is a variable that can take on a countable number of possible values along a specified interval.

Continuous Probability Distributions A continuous random variable is a variable that can take on any of the possible values between two points.

Examples of Continuous Random variables • Time required to perform a job • Financial ratios • Product weights • Volume of soft drink in a 12-ounce can • Interest rates • Income levels • Distance between two points

Continuous Probability Distributions The probability distribution of a continuous random variable is represented by a probability density function that defines a curve.

Continuous Probability Distributions (a) Discrete Probability Distribution (b) Probability Density Function P(x) f(x) x x Possible Values of x Possible Values of x

Normal Probability Distribution The Normal Distribution is a bell-shaped, continuous distribution with the following properties: 1. It is unimodal. 2. It is symmetrical; this means 50% of the area under the curve lies left of the center and 50% lies right of center. 3. The mean, median, and mode are equal. 4. It is asymptotic to the x-axis. 5. The amount of variation in the random variable determines the width of the normal distribution.

Normal Probability Distribution NORMAL DISTRIBUTION DENSITY FUNCTION where: x = Any value of the continuous random variable  = Population standard deviation e = Base of the natural log = 2.7183  = Population mean

Normal Probability Distribution (Figure 5-2) f(x) x  Mean Median Mode

Differences Between Normal Distributions (Figure 5-3) x (a) x (b) x (c)

Standard Normal Distribution The standard normal distribution is a normal distribution which has a mean = 0.0 and a standard deviation = 1.0. The horizontal axis is scaled in standardized z-values that measure the number of standard deviations a point is from the mean. Values above the mean have positive z-values and those below have negative z-values.

Standard Normal Distribution STANDARDIZED NORMAL Z-VALUE where: x = Any point on the horizontal axis  = Standard deviation of the normal distribution  = Population mean z = Scaled value (the number of standard deviations a point x is from the mean)

Areas Under the Standard Normal Curve (Using Table 5-1) 0.1985 X 0.52 Example: z = 0.52 (or -0.52) P(0 < z < .52) = 0.1985 or 19.85%

Areas Under the Standard Normal Curve (Table 5-1)

Standard Normal Example (Figure 5-6) Probabilities from the Normal Curve for Westex 0.1915 0.50 x z x=45 50 z= -.50

Standard Normal Example (Figure 5-7) z z=-1.25 x=7.5 From the normal table: P(-1.25  z  0) = 0.3944 Then, P(x  7.5 hours) = 0.50 - 0.3944 = 0.1056

Uniform Probability Distribution The uniform distribution is a probability distribution in which the probability of a value occurring between two points, a and b, is the same as the probability between any other two points, c and d, given that the distribution between a and b is equal to the distance between c and d.

Uniform Probability Distribution CONTINUOUS UNIFORM DISTRIBUTION where: f(x) = Value of the density function at any x value a = Lower limit of the interval from a to b b = Upper limit of the interval from a to b

Uniform Probability Distributions (Figure 5-16) f(x) f(x) for 2  x  5 for 3  x  8 .50 .50 .25 .25 2 5 3 8 a b a b

Exponential Probability Distribution The exponential probability distribution is a continuous distribution that is used to measure the time that elapses between two occurrences of an event.

Exponential Probability Distribution EXPONENTIAL DISTRIBUTION A continuous random variable that is exponentially distributed has the probability density function given by: where: e = 2.71828. . . 1/ = The mean time between events ( >0)

Exponential Distributions (Figure 5-18) Lambda = 3.0 (Mean = 0.333) f(x) Lambda = 2.0 (Mean = 0.5) Lambda = 1.0 (Mean = 1.0) Lambda = 0.50 (Mean = 020) x Values of x

Exponential Probability

Key Terms • Continuous Random Variable • Discrete Random Variable • Exponential Distribution • Normal Distribution • Standard Normal Distribution Standard Normal Table • Uniform Distribution • z-Value